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The Quarterly Journal of Mathematics
, Volume Advance Article – Mar 12, 2018

19 pages

/lp/ou_press/the-singularity-category-of-a-quadratic-monomial-algebra-gtV9pr5FFS

- Publisher
- Oxford University Press
- Copyright
- © The Author(s) 2018. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com
- ISSN
- 0033-5606
- eISSN
- 1464-3847
- D.O.I.
- 10.1093/qmath/hay006
- Publisher site
- See Article on Publisher Site

Abstract We exploit singular equivalences between artin algebras that are induced from certain functors between the stable module categories. Such functors are called pre-triangle equivalences. We construct two pre-triangle equivalences connecting the stable module category over a quadratic monomial algebra to the one over an algebra with radical square zero. Consequently, we obtain an explicit singular equivalence between the two algebras. It turns out that this singular equivalence restricts to a triangle equivalence between their stable categories of Gorenstein-projective modules, and thus induces a triangle equivalence between their Gorenstein defect categories. 1. Introduction Let A be an artin algebra. The singularity category Dsg(A) of A is introduced in [7] under the name ‘the stable derived category’. The terminology is justified by the following fact: the algebra A has finite global dimension if and only if the singularity category Dsg(A) is trivial. Hence, the singularity category provides a homological invariant for algebras of infinite global dimension. The singularity category captures the stable homological property of an algebra. More precisely, certain information of the syzygy endofunctor on the stable A-module category is encoded in Dsg(A). Indeed, as observed in [21], the singularity category is equivalent to the stabilization of the pair that consists of the stable module category and the syzygy endofunctor on it; see also [4]. This fact is used in [10] to describe the singularity category of an algebra with radical square zero. We mention that related results appear in [19, 26]. By the fundamental result in [7], the stable category of Gorenstein-projective A-modules might be viewed as a triangulated subcategory of Dsg(A). Moreover, if the algebra A is Gorenstein, the two categories are triangle equivalent. We mention that the study of Gorenstein-projective modules goes back to [2] under the name ‘modules of G-dimension zero’. The Verdier quotient triangulated category Ddef(A) of Dsg(A) by the stable category of Gorenstein-projective A-modules is called the Gorenstein defect category of A in [6]. This terminology is justified by the fact that the algebra A is Gorenstein if and only if the category Ddef(A) is trivial. In other words, the Gorenstein defect category measures how far the algebra is from being Gorenstein. By a singular equivalence between two algebras, we mean a triangle equivalence between their singularity categories. We observe that a derived equivalence implies a singular equivalence. However, the converse is not true; for such examples, see [9, 23]. In general, a singular equivalence does not induce a triangle equivalence between Gorenstein defect categories. We mention the work [30], where a class of nice singular equivalences are studied. The aim of this paper is to study the singularity category of a quadratic monomial algebra. The main ingredient is the following observation: for two algebras, a certain functor between their stable module categories induces a singular equivalence after the stabilization. We call such a functor a pre-triangle equivalence between the stable module categories. More generally, the two stable module categories are called pre-triangle quasi-equivalent provided that there is a zigzag of pre-triangle equivalences connecting them. In this case, we also have a singular equivalence. The main result Theorem 4.5 claims a pre-triangle quasi-equivalence between the stable module category of a quadratic monomial algebra and the one of an algebra with radical square zero. Combining this with the results in [10, 13, 27], we describe the singularity category of a quadratic monomial algebra via the category of finitely generated graded projective modules over the Leavitt path algebra of a certain quiver; see Proposition 5.3. We mention that this description extends the result in [20] on the singularity category of a gentle algebra; see also [8, 12]. The paper is organized as follows. In Section 2, we recall the stabilization of a looped category. We introduce the notion of a pre-stable equivalence between looped categories, which is a functor between looped categories that induces an equivalence after the stabilization. A pre-stable equivalence in the left triangulated case is called a pre-triangle equivalence, which induces a triangle equivalence after the stabilization. In Section 3, we recall the result in [21] which states that the singularity category of an algebra is triangle equivalent to the stabilization of the stable module category. Therefore, a pre-triangle equivalence between stable module categories induces a singular equivalence; see Proposition 3.2 and compare Proposition 3.6. We include explicit examples of pre-triangle equivalences between stable module categories. In Section 4, we associate an algebra B with radical square zero to a quadratic monomial algebra A; compare [12]. We construct explicitly two pre-triangle equivalences connecting the stable A-module category to the stable B-module category. Then we obtain the required singular equivalence between A and B; see Theorem 4.5. In Section 5, we combine Theorem 4.5 with the results in [10, 13, 27] on the singularity category of an algebra with radical square zero. We describe the singularity category and the Gorenstein defect category of a quadratic monomial algebra via the categories of finitely generated graded projective modules over Leavitt path algebras of certain quivers; see Proposition 5.3. We discuss some concrete examples at the end. 2. The stabilization of a looped category In this section, we recall the construction of the stabilization of a looped category. The basic references are [16, Chapter I], [28, Section 1], [21] and [4, Section 3]. Following [4], a looped category (C,Ω) consists of a category C with an endofunctor Ω:C→C, called the loop functor. The looped category (C,Ω) is said to be stable if the loop functor Ω is an auto-equivalence on C, while it is strictly stable if Ω is an automorphism. By a looped functor (F,δ) between two looped categories (C,Ω) and (D,Δ), we mean a functor F:C→D together with a natural isomorphism δ:FΩ→ΔF. For a looped functor (F,δ), we define inductively for each i≥1 a natural isomorphism δi:FΩi→ΔiF such that δ1=δ and δi+1=Δiδ◦δiΩ. Set δ0 to be the identity transformation on F, where Ω0 and Δ0 are defined to be the identity functors. We say that a looped functor (F,δ):(C,Ω)→(D,Δ) is strictly looped provided that FΩ=ΔF as functors and δ is the identity transformation on FΩ. In this case, we write (F,δ) as F; compare [16, 1.1]. Let (C,Ω) be a looped category. We define a category S=S(C,Ω) as follows. The objects of S are pairs (X,n) with X an object in C and n∈Z. The Hom-set is defined by the following formula: HomS((X,n),(Y,m))=colimHomC(Ωi−n(X),Ωi−m(Y)), (2.1) where i runs over all integers satisfying i≥n and i≥m. An element f in HomS((X,n),(Y,m)) is said to have an ith representative fi:Ωi−n(X)→Ωi−m(Y) provided that the canonical image of fi equals f. The composition of morphisms in S is induced by the one in C. We observe that Ω˜:S→S sending (X,n) to (X,n−1) is an automorphism. Then we have a strictly stable category (S,Ω˜). There is a canonical functor S:C→S sending X to (X,0), and a morphism f to S(f) whose 0th representative is f. For an object X in C, we have a natural isomorphism θX:(ΩX,0)⟶(X,−1), whose 0th representative is IdΩX. Indeed, this yields a looped functor (S,θ):(C,Ω)⟶(S,Ω˜). This process is called in [16] the stabilization of the looped functor (C,Ω). We mention that S:C→S is an equivalence if and only if (C,Ω) is a stable category, in which case we identify (C,Ω) with (S,Ω˜). The stabilization functor (S,θ) enjoys a universal property; see [16, Proposition 1.1]. Let (F,δ):(C,Ω)→(D,Δ) be a looped functor with (D,Δ) a strictly stable category. We denote by Δ−1 the inverse of Δ. Then there is a unique functor F˜:(S,Ω˜)→(D,Δ) which is strictly looped satisfying F=F˜S and δ=F˜θ. The functor F˜ sends (X,n) to Δ−nF(X). For a morphism f:(X,n)→(Y,m) whose ith representative is given by fi:Ωi−n(X)→Ωi−m(Y), we have F˜(f)=Δ−i((δYi−m)◦F(fi)◦(δXi−n)−1):Δ−nF(X)⟶Δ−mF(Y). (2.2) Lemma 2.1 Keep the notation as above. Then the functor F˜:(S,Ω˜)→(D,Δ)is an equivalence if and only if the following conditions are satisfied: for any morphism g:F(X)→F(Y)in D, there exist i≥0and a morphism f:Ωi(X)→Ωi(Y)in Csatisfying Δi(g)=δYi◦F(f)◦(δXi)−1; for any two morphisms f,f′:X→Yin Cwith F(f)=F(f′), there exists i≥0such that Ωi(f)=Ωi(f′); for any object Zin D, there exist i≥0and an object Xin Csatisfying Δi(Z)≃F(X). Proof Indeed, the above three conditions are equivalent to the statements that F˜ is full, faithful and dense, respectively. We refer to [28, 1.2 Proposition] for the details and compare [4, Proposition 3.4].□ We now apply Lemma 2.1 to a specific situation. Let (F,δ):(C,Ω)→(C′,Ω′) be a looped functor. Consider the composition (C,Ω)⟶(F,δ)(C′,Ω′)⟶(S,θ)(S(C′,Ω′),Ω˜′). (2.3) By the universal property of the stabilization, there is a unique strictly looped functor S(F,δ):(S(C,Ω),Ω˜)→(S(C′,Ω′),Ω˜′) making the following diagram commutative: We call the functor S(F,δ) the stabilization of (F,δ). Proposition 2.2 Let (F,δ):(C,Ω)→(C′,Ω′)be a looped functor. Then its stabilization S(F,δ)is an equivalence if and only if the following conditions are satisfied: (S1) for any morphism g:F(X)→F(Y)in C′, there exist i≥0and a morphism f:Ωi(X)→Ωi(Y)in Csatisfying Ω′i(g)=δYi◦F(f)◦(δXi)−1; (S2) for any two morphisms f,f′:X→Yin Cwith F(f)=F(f′), there exists i≥0such that Ωi(f)=Ωi(f′); (S3) for any object C′in C′, there exist i≥0and an object Xin Csatisfying Ω′i(C′)≃F(X). The looped functor (F,δ) is called a pre-stable equivalence if it satisfies (S1)–(S3). The result implies that a pre-stable equivalence induces an equivalence between the stabilized categories. Proof Write (D,Δ)=(S(C′,Ω′),Ω˜′) and F˜=S(F,δ). Write the composition (2.3) as (SF,∂). Then for an object X in C, the morphism ∂X:SFΩ(X)→Ω˜′SF(X) equals θFX◦S(δX). We make the following observation: for a morphism f:Ωl(X)→Ωl(Y) in C, the morphism ∂Yl◦SF(f)◦(∂Xl)−1 has a 0th representative δYl◦F(f)◦(δXl)−1. We claim that for each 1≤i≤3, the condition ( Si) for (F,δ) is equivalent to the condition ( i) in Lemma 2.1 for (SF,∂). Then we are done by Lemma 2.1. In what follows, we only prove that ( Si) implies ( i). By reversing the argument, we obtain the converse implication. Assume that (S1) for (F,δ) holds. We take a morphism g:SF(X)=(FX,0)→SF(Y)=(FY,0) in D. We assume that g has a jth representative gj:Ω′jF(X)→Ω′jF(Y). Consider the morphism h:F(ΩjX)→F(ΩjY) by h=(δYj)−1◦gj◦δXj. Then by (S 1), there exist i≥0 and a morphism f:Ωi+j(X)→Ωi+j(Y) satisfying Ω′i(h)=(δΩjYi)◦F(f)◦(δΩjXi)−1. Then we have Δi+j(g)=∂Yi+j◦SF(f)◦(∂Xi+j)−1. Here, we use the observation above and the fact that Δi+j(g) has a 0th representative Ω′i(gj). The we have (1) for (SF,∂). Assume that (S2) for (F,δ) holds. We take two morphisms f,f′:X→Y in C with SF(f)=SF(f′). Then there exists j≥0 such that Ω′jF(f)=Ω′jF(f′). Using the natural isomorphism δj, we infer that FΩj(f)=FΩj(f′). By (S2), there exists i≥0 such that Ωi+j(f)=Ωi+j(f′), proving (2) for (SF,∂). Assume that (S3) for (F,δ) holds. We take any object (C′,n) in (D,Δ). We may assume that n≥0. Otherwise, we use the isomorphism θC′−n:((Ω′)−n(C′),0)≃(C′,n). By (S3), there exist j≥0 and an object X in C satisfying Ω′j(C′)≃F(X). We observe that Δj+n(C′,n)=(C′,−j), which is isomorphic to SΩ′j(C′), which is further isomorphic to SF(X). Set i=j+n. Then we have the required isomorphism Δi(C′,n)≃SF(X) in (3) for (SF,∂). This completes the proof of the claim.□ We make an easy observation. Corollary 2.3 Let (F,δ):(C,Ω)→(C′,Ω′)be a looped functor. Assume that Fis fully faithful. Then (F,δ)is a pre-stable equivalence if and only if (S3) holds. Proof By the fully-faithfulness of F, the conditions (S1) and (S2) hold trivially. We just take i=0 in both the conditions.□ We say that two looped categories (C,Ω) and (C′,Ω′) are pre-stably quasi-equivalent provided that there exists a chain of looped categories (C,Ω)=(C1,Ω1),(C2,Ω2),…,(Cn,Ωn)=(C′,Ω′) (2.4) such that for each 1≤i≤n−1, there exists a pre-stable equivalence from (Ci,Ωi) to (Ci+1,Ωi+1), or a pre-stable equivalence from (Ci+1,Ωi+1) to (Ci,Ωi). We have the following immediate consequence of Proposition 2.2. Corollary 2.4 Let (C,Ω)and (C′,Ω′)be two looped categories which are pre-stably quasi-equivalent. Then there is a looped functor (S(C,Ω),Ω˜)⟶∼(S(C′,Ω′),Ω˜′),which is an equivalence.□ Let (F,δ):(C,Ω)→(C′,Ω′) be a looped functor. A full subcategory X⊆C is said to be saturated provided that the following conditions are satisfied: (Sa1) For each object X in C, there is a morphism ηX:X→G(X) with G(X) in X such that F(ηX) is an isomorphism and that Ωd(ηX) is an isomorphism for some d≥0. (Sa2) For a morphism f:X→Y, there is a morphism G(f):G(X)→G(Y) with G(f)◦ηX=ηY◦f. (Sa3) The conditions (S1)–(S3) above hold by requiring that all the objects X,Y belong to X. Example 2.5 Let (F,δ):(C,Ω)→(C′,Ω′) be a looped functor. Assume that F has a right adjoint functor G, which is fully faithful. Assume further that the unit η:IdC→GF satisfies the following condition: for each object X, there exists d≥0 with Ωd(ηX) an isomorphism. Take X to be the essential image of G. We claim that X⊆C is a saturated subcategory. Indeed, the restriction F∣X:X→C′ is an equivalence. Then (Sa3) holds trivially, by taking i to be zero in (S1)–(S3). The conditions (Sa1) and (Sa2) are immediate from the assumption. Here, we use the well-known fact that F(η) is a natural isomorphism, since G is fully faithful. Lemma 2.6 Let (F,δ):(C,Ω)→(C′,Ω′)be a looped functor, and X⊆Ca saturated subcategory. Then the conditions (S1)–(S3) hold, that is, the functor (F,δ)is a pre-stable equivalence. Proof It suffices to verify (S1) and (S2). For (S1), take any morphism g:F(X)→F(Y) in C′. Consider g′=F(ηY)◦g◦F(ηX)−1:FG(X)→FG(Y). Then by (Sa3), there exist i≥0 and f′:Ωi(GX)→Ωi(GY) with Ω′i(g′)=δGYi◦F(f′)◦(δGXi)−1. We may assume that i is large enough such that both Ωi(ηX) and Ωi(ηY) are isomorphisms. Take f=(Ωi(ηY))−1◦f′◦Ωi(ηX), which is the required morphism in (S1). Let f,f′:X→Y be morphisms with F(f)=F(f′). Applying (Sa2) and using the isomorphisms F(ηX) and F(ηY), we have FG(f)=FG(f′). By (Sa3), we have ΩiG(f)=ΩiG(f′) for some i≥0. We assume that i is large enough such that both Ωi(ηX) and Ωi(ηY) are isomorphisms. Then we infer from (Sa2) that Ωi(f)=Ωi(f′). We are done with (S2).□ We will specialize the consideration to left triangulated categories. A looped category (C,Ω) is additive provided that C is an additive category and the loop functor Ω is an additive functor. We recall that a left triangulated category (C,Ω,E) consists of an additive looped category (C,Ω) and a class E of left triangles in C satisfying certain axioms, which are analogous to those for a triangulated category, but the endofunctor Ω is possibly not an auto-equivalence. The following convention is usual. We call a left triangulated category (C,Ω,E) a triangulated category, provided that the category (C,Ω) is stable, that is, the endofunctor Ω is an auto-equivalence. In this case, the translation functor Σ of C is a quasi-inverse of Ω. Then this notion is equivalent to the original one of a triangulated category in the sense of Verdier. For details, we refer to [5] and compare [21]. In what follows, we write C for the left triangulated category (C,Ω,E). A looped functor (F,δ) between two left triangulated categories C and C′=(C′,Ω′,E′) is called a triangle functor if F is an additive functor and sends left triangles to left triangles. We sometimes suppress the natural isomorphism δ and simply denote the triangle functor by F. A triangle functor which is a pre-stable equivalence is called a pre-triangle equivalence. Two left triangulated categories C and C′ are pre-triangle quasi-equivalent if they are pre-stably quasi-equivalent such that all the categories in (2.4) are left triangulated and all the pre-stable equivalences connecting them are pre-triangle equivalences. For a left triangulated category C=(C,Ω,E), the stabilized category S(C)≔(S(C,Ω),Ω˜,E˜) is a triangulated category, where the translation functor Σ=(Ω˜)−1 and the triangles in E˜ are induced by the left triangles in E; see [4, Section 3]. Corollary 2.7 Let Cand C′be two left triangulated categories which are pre-triangle quasi-equivalent. Then there is a triangle equivalence S(C)→∼S(C′).□ 3. The singularity categories and singular equivalences In this section, we recall the notion of the singularity category of an algebra. We shall show that for two algebras whose stable module categories are pre-triangle quasi-equivalent, their singularity categories are triangle equivalent; see Proposition 3.2 and compare Proposition 3.6 below. Let k be a commutative artinian ring with a unit. We emphasize that all the functors and categories are required to be k-linear in this section. Let A be an artin k-algebra. We denote by A-mod the category of finitely generated left A-modules, and by A-proj the full subcategory consisting of projective modules. We denote by A-mod̲ the stable category of A-mod modulo projective modules [3, p. 104]. The morphism space Hom̲A(M,N) of two modules M and N in A-mod̲ is defined to be HomA(M,N)/p(M,N), where p(M,N) denotes the k-submodule formed by morphisms that factor through projective modules. For a morphism f:M→N, we write f¯ for its image in Hom̲A(M,N). Recall that for an A-module M, its syzygy ΩA(M) is the kernel of its projective cover P(M)→pMM. We fix for M a short exact sequence 0→ΩA(M)→iMP(M)→pMM→0. This gives rise to the syzygy functor ΩA:A-mod̲→A-mod̲; see [3, p. 124]. Indeed, A-mod̲≔(A-mod̲,ΩA,EA) is a left triangulated category, where EA consists of left triangles that are induced from short exact sequences in A-mod. More precisely, given a short exact sequence 0→X→fY→gZ→0, we have the following commutative diagram: Then ΩA(Z)→h¯X→f¯Y→g¯Z is a left triangle in EA. As recalled in Section 2, the stabilized category S(A-mod̲) is a triangulated category. There is a more well-known description of this stabilized category as the singularity category; see [21]. To recall this, we denote by Db(A-mod) the bounded derived category of A-mod. We identify an A-module M with the corresponding stalk complex concentrated at degree zero, which is also denoted by M. Recall that a complex in Db(A-mod) is perfect provided that it is isomorphic to a bounded complex consisting of projective modules. The full subcategory consisting of perfect complexes is denoted by perf(A), which is a triangulated subcategory of Db(A-mod) and is closed under direct summands; see [7, Lemma 1.2.1]. Following [22], the singularity category of an algebra A is defined to be the Verdier quotient triangulated category Dsg(A)=Db(A-mod)/perf(A); compare [7, 15, 21]. We denote by q:Db(A-mod)→Dsg(A) the quotient functor. We denote a complex of A-modules by X•=(Xn,dn)n∈Z, where Xn are A-modules and the differentials dn:Xn→Xn+1 are homomorphisms of modules satisfying dn+1◦dn=0. The translation functor Σ both on Db(A-mod) and Dsg(A) sends a complex X• to a complex Σ(X•), which is given by Σ(X)n=Xn+1 and dΣXn=−dXn+1. Consider the following functor: FA:A-mod̲⟶Dsg(A) sending a module M to the corresponding stalk complex concentrated at degree zero, and a morphism f¯ to q(f). Here, the well-definedness of FA on morphisms is due to the fact that a projective module is isomorphic to the zero object in Dsg(A). For an A-module M, we consider the two-term complex C(M)=⋯→0→P(M)→pMM→0→⋯ with P(M) at degree zero. Then we have a quasi-isomorphism iM:ΩA(M)→C(M). The canonical inclusion canM:Σ−1(M)→C(M) becomes an isomorphism in Dsg(A). Then we have a natural isomorphism δM=q(canM)−1◦q(iM):FAΩA(M)⟶Σ−1FA(M). In other words, (FA,δ):(A-mod̲,ΩA)→(Dsg(A),Σ−1) is a looped functor. Indeed, FA is an additive functor and sends left triangles to (left) triangles. Then we have a triangle functor (FA,δ):A-mod̲⟶Dsg(A). Applying the universal property of the stabilization to (FA,δ), we obtain a strictly looped functor F˜A:S(A-mod̲)⟶Dsg(A), which is also a triangle functor; see [4, 3.1]. The following basic result is due to [21]. For a detailed proof, we refer to [4, Corollary 3.9]. Lemma 3.1 Keep the notation as above. Then F˜A:S(A-mod̲)→Dsg(A)is a triangle equivalence. By a singular equivalence between two algebras A and B, we mean a triangle equivalence between their singularity categories. Proposition 3.2 Let Aand Bbe two artin algebras. Assume that the stable categories A-mod̲and B-mod̲are pre-triangle quasi-equivalent. Then there is a singular equivalence between Aand B. Proof We just combine Lemma 3.1 and Corollary 2.7.□ In the following two examples, pre-triangle equivalences between stable module categories are explicitly given. We require that k acts centrally on any bimodules. Example 3.3 Let A and B′ be artin algebras, and let MB′A be an A- B′-bimodule. Consider the upper triangular matrix algebra B=(A0MB′). We recall that a left B-module is a column vector (XY), where XA and YB′ are a left A-module and a left B′-module with an A-module homomorphism ϕ:M⊗B′Y→X, respectively; compare [3, III]. We call ϕ the structure morphism of the B-module (XY). Consider the natural full embedding i:A-mod→B-mod, sending an A-module X to i(X)=(X0). Since i preserves projective modules and is exact, it commutes with taking the syzygies. Then we have the induced functor i:A-mod̲→B-mod̲, which is a triangle functor. We claim that the induced functor i is a pre-triangle equivalence if and only if the algebra B′ has finite global dimension. In this case, by Proposition 3.2, there is a triangle equivalence Dsg(A)→∼Dsg(B); compare [9, Theorem 4.1(1)]. Indeed, the induced functor i is fully faithful. By Corollary 2.3, we only need to consider the condition (S3). Then we are done by the following fact: for any B-module (XY) and d≥0, we have ΩBd(XY)=(X′ΩB′d(Y)) for some A-module X′. Hence, if ΩB′d(Y)=0, the B-module ΩBd(XY) lies in the essential image of i. The following example is somehow more difficult. Example 3.4 Let A and B′ be artin algebras, and let NAB′ be an A- B′-bimodule. Consider the upper triangular matrix algebra B=(B′0NA). We assume that B′ has finite global dimension. Consider the natural projection functor p:B-mod→A-mod, sending a B-module (XY) to the A-module Y. It is an exact functor which sends projective modules to projective modules. Then we have the induced functor p:B-mod̲→A-mod̲, which is a triangle functor. For an A-module Y, (0Y) is naturally a B-module with the zero structure morphism N⊗AY→0. Take X to be the full subcategory of B-mod̲ consisting of modules of the form (0Y). We claim that X is a saturated subcategory of B-mod̲. Then by Lemma 2.6, the induced functor p is a pre-triangle equivalence. Therefore, by Proposition 3.2, there is a triangle equivalence Dsg(B)→∼Dsg(A); compare [9, Theorem 4.1(2)]. We now prove the claim. For a B-module C=(XY), we consider the projection ηC:(XY)→G(C)=(0Y). Since its kernel has finite projective dimension, it follows that ΩBd(ηC) is an isomorphism for d large enough. We observe that p(ηC) is an isomorphism. Then we have (Sa1). The conditions (Sa2) and (Sa3) are trivial. Here for (S2) in X, we use the following fact: if a morphism f:Y→Y′ of A-module factors through a projective A-module P, then the morphism (0f):(0Y)→(0Y′) of B-modules factors though (0P), which has finite projective dimension; consequently, we have ΩBd(0f)=0 for d large enough. Let M be a left A-module. Then M*=HomA(M,A) is a right A-module. Recall that an A-module M is Gorenstein-projective provided that there is an acyclic complex P• of projective A-modules such that the Hom-complex (P•)*=HomA(P•,A) is still acyclic and that M is isomorphic to a certain cocycle Zi(P•) of P•. We denote by A-Gproj the full subcategory of A-mod formed by Gorenstein-projective A-modules. We observe that A-proj⊆A-Gproj. We recall that the full subcategory A-Gproj⊆A-mod is closed under direct summands, kernels of epimorphisms and extensions; compare [2, (3.11)]. In particular, for a Gorenstein-projective A-module M all its syzygies ΩAi(M) are Gorenstein-projective. Since A-Gproj⊆A-mod is closed under extensions, it becomes naturally an exact category in the sense of Quillen [24]. Moreover, it is a Frobenius category, that is, it has enough (relatively) projective and enough (relatively) injective objects, and the class of projective objects coincides with the class of injective objects. In fact, the class of projective-injective objects in A-Gproj equals A-proj. For details, we compare [4, Proposition 2.13]. We denote by A-Gproj̲ the full subcategory of A-mod̲ consisting of Gorenstein-projective A-modules. Then the syzygy functor ΩA restricts to an auto-equivalence ΩA:A-Gproj̲→A-Gproj̲. Moreover, the stable category A-Gproj̲ becomes a triangulated category such that the translation functor is given by a quasi-inverse of ΩA, and that the triangles are induced by short exact sequences in A-Gproj. These are consequences of a general result in [14, Chapter I.2]. The inclusion functor inc:A-Gproj̲→A-mod̲ is a triangle functor between left triangulated categories. We consider the composite of triangle functors GA:A-Gproj̲⟶incA-mod̲⟶FADsg(A). Let M,N be Gorenstein-projective A-modules. By the fully-faithfulness of the functor ΩA:A-Gproj̲→A-Gproj̲, the natural map Hom̲A(M,N)⟶HomS(A-mod̲)(M,N) induced by the stabilization functor S:A-mod̲→S(A-mod̲) is an isomorphism. We identify S(A-mod̲) with Dsg(A) by Lemma 3.1. Then this isomorphism implies that the triangle functor GA is fully faithful; compare [7, Theorem 4.1] and [15, Theorem 4.6]. Recall from [7, 15] that an artin algebra A is Gorenstein if the regular module A has finite injective dimension on both sides. Indeed, the two injective dimensions are equal. We mention that a self-injective algebra is Gorenstein, where any module is Gorenstein-projective. The following result is also known. As a consequence, for a self-injective algebra A the stable module category A-mod̲ and Dsg(A) are triangle equivalent; see [21] and [25, Theorem 2.1]. Lemma 3.5 Let Abe an artin algebra. Then the following statements are equivalent: The algebra Ais Gorenstein. The inclusion functor inc:A-Gproj̲→A-mod̲is a pre-triangle equivalence. The functor GA:A-Gproj̲→Dsg(A)is a triangle equivalence. Proof Recall that A is Gorenstein if and only if for any module X, there exists d≥0 with ΩAd(X) Gorenstein-projective; see [18]. The inclusion functor in (2) is fully faithful. By Corollary 2.3, it is a pre-triangle equivalence if and only if the condition (S3) in A-mod̲ is satisfied. Then the equivalence ‘ (1)⇔(2)’ follows. Since ΩA:A-Gproj̲→A-Gproj̲ is an auto-equivalence, we identify A-Gproj̲ with its stabilization S(A-Gproj̲). By Lemma 3.1, we identify Dsg(A) with S(A-mod̲). Then the functor GA is identified with the stabilization of the inclusion functor in (2). Then the equivalence ‘ (2)⇔(3)’ follows from Proposition 2.2. □ Recall from [6] that the Gorenstein defect category of an algebra A is defined to be the Verdier quotient triangulated category Ddef(A)=Dsg(A)/ImGA, where ImGA denotes the essential image of the fully-faithful triangle functor GA, and thus is a triangulated subcategory of Dsg(A). By Lemma 3.5(3), the algebra A is Gorenstein if and only if Ddef(A) is trivial; see also [6]. The following observation implies that pre-triangle equivalences seem to be ubiquitous in the study of singular equivalences; compare Proposition 3.2. Proposition 3.6 Let Aand Bbe artin algebras. Assume that Bis a Gorenstein algebra and that there is a singular equivalence between Aand B. Then there is a pre-triangle equivalence from A-mod̲to B-mod̲. Proof Using the triangle equivalence GB, we obtain a triangle equivalence H:Dsg(A)⟶B-Gproj̲. More precisely, we have H=GB−1L, where L:Dsg(A)→Dsg(B) is the assumed singular equivalence and GB−1 is a quasi-inverse of GB. Then we have the following composite of triangle functors: F:A-mod̲⟶FADsg(A)⟶HB-Gproj̲⟶incB-mod̲. We claim that F is a pre-triangle equivalence. Indeed, the functor FA is a pre-triangle equivalence by Lemma 3.1, where we identify Dsg(A) with its stabilization S(Dsg(A)). The inclusion functor is a pre-triangle equivalence by Lemma 3.5(2). Therefore, all the three functors above are pre-triangle equivalences. Then as their composition, so is the functor F.□ 4. The singularity category of a quadratic monomial algebra In this section, we study the singularity category of a quadratic monomial algebra A. We consider the algebra B with radical square zero that is defined by the relation quiver of A. The main result claims that there is a pre-triangle quasi-equivalence between the stable A-module category and the stable B-module category. Consequently, we obtain an explicit singular equivalence between A and B. For the ease of the reader, we recall some notation on quivers and quadratic monomial algebras. Let Q=(Q0,Q1;s,t) be a finite quiver, where Q0 is the set of vertices, Q1 the set of arrows, and s,t:Q1→Q0 are maps which assign to each arrow α its starting vertex s(α) and its terminating vertex t(α). A path p of length n in Q is a sequence p=αn⋯α2α1 of arrows such that s(αi)=t(αi−1) for 2≤i≤n; moreover, we define its starting vertex s(p)=s(α1) and its terminating vertex t(p)=t(αn). We observe that a path of length one is just an arrow. To each vertex i, we associate a trivial path ei of length zero, and set s(ei)=i=t(ei). For two paths p and q with s(p)=t(q), we write pq for their concatenation. As convention, we have p=pes(p)=et(p)p. For two paths p and q in Q, we say that q is a sub-path of p provided that p=p″qp′ for some paths p″ and p′. Let k be a field. The path algebra kQ of a finite quiver Q is defined as follows. As a k-vector space, it has a basis given by all the paths in Q. For two paths p and q, their multiplication is given by the concatenation pq if s(p)=t(q), and it is zero, otherwise. The unit of kQ equals ∑i∈Q0ei. Denote by J the two-sided ideal of kQ generated by arrows. Then Jd is spanned by all the paths of length at least d for each d≥2. A two-sided ideal I of kQ is admissible provided that Jd⊆I⊆J2 for some d≥2. In this case, the quotient algebra A=kQ/I is finite-dimensional. We recall that an admissible ideal I of kQ is quadratic monomial provided that it is generated by some paths of length two. In this case, the quotient algebra A=kQ/I is called a quadratic monomial algebra. Observe that the algebra A is with radical square zero if and only if I=J2. We call kQ/J2 the algebra with radical square zero defined by the quiver Q. In what follows, A=kQ/I is a quadratic monomial algebra. We denote by F the set of paths of length two contained in I. Following [29], a path p in Q is non-zero in A provided that it does not belong to I, or equivalently, p does not contain a sub-path in F. In this case, we abuse the image p+I in A with p. The set of non-zero paths forms a k-basis for A. For a path p in I, we write p=0 in A. For a non-zero path p, we consider the left ideal Ap generated by p, which has a k-basis given by the non-zero paths q such that q=q′p for some path q′. We observe that for a vertex i, Aei is an indecomposable projective A-module. Then we have a projective cover πp:Aet(p)→Ap sending et(p) to p. Lemma 4.1 Let A=kQ/Ibe a quadratic monomial algebra. Then the following statements hold: For a non-zero path p=αp′with αan arrow, there is an isomorphism Ap≃Aαof A-modules sending xpto xαfor any path xwith s(x)=t(p). For an arrow α, we have a short exact sequence of A-modules 0⟶⨁{β∈Q1∣βα∈F}Aβ⟶incAet(α)⟶παAα⟶0, (4.1)where ‘ inc’ denotes the inclusion map. For any A-module M, there is an isomorphism ΩA2(M)≃⨁α∈Q1(Aα)nαfor some integers nα. Proof (1) is trivial and (2) is straightforward; compare the first paragraph in [29, p. 162]. In view of (1), the statement (3) is a special case of [29, Theorem I].□ Let α be an arrow such that the set {β∈Q1∣βα∈F} is non-empty. By (4.1), this is equivalent to the condition that the A-module Aα is non-projective. Denote by N(α)={α′∈Q1∣t(α′)=t(α),βα′∈Fforeacharrowβsatisfyingβα∈F}. Set Z(α)=⨁α′∈N(α)α′A, which is the right ideal generated by N(α). We observe that α∈N(α). The second statement of the following result is analogous to [12, Lemma 2.3]. Lemma 4.2 Let α,α′be two arrows. We assume that the set {β∈Q1∣βα∈F}is non-empty. Then we have the following statements: There is an isomorphism HomA(Aα,A)→Z(α)sending fto f(α). There is a k-linear isomorphism Hom̲A(Aα,Aα′)=Z(α)∩Aα′Z(α)α′. (4.2) If α′does not belong to N(α), we have Hom̲A(Aα,Aα′)=0. If α′belongs to N(α), there is a unique epimorphism π=πα,α′:Aα→Aα′sending αto α′and Hom̲A(Aα,Aα′)=kπ¯. Proof We observe that Z(α) has a k-basis given by non-zero paths q which satisfy t(q)=t(α) and βq=0 for each arrow β with βα∈F. Then we infer (1) by applying HomA(−,A) to (4.1) and using the canonical isomorphism HomA(Aet(α),A)≃et(α)A. For (2), we identify for each left ideal K of A, HomA(Aα,K) with the subspace of HomA(Aα,A) formed by those morphisms whose image is contained in K. Therefore, we identify HomA(Aα,Aα′) with Z(α)∩Aα′, HomA(Aα,Aet(α′)) with Z(α)∩Aet(α′). Recall the projective cover πα′:Aet(α′)→Aα′. The subspace p(Aα,Aα′) formed by those morphisms factoring through projective modules equals the image of the map HomA(πα′,A). This image is then identified with Z(α)α′. Then the required isomorphism follows. The statement (3) is an immediate consequence of (2), since in this case we have Z(α)∩Aα′=Z(α)α′. For (4), we observe in this case that Z(α)∩Aα′=(Z(α)α′)⊕kα′. It follows from (3) that Hom̲A(Aα,Aα′) is one dimensional. The existence of the surjective homomorphism π is by the isomorphism in (1), under which π corresponds to the element α′. Then we are done.□ Remark 4.3 Assume that α′∈N(α). In particular, t(α)=t(α′). Then we have the following commutative diagram: The leftmost inclusion uses the fact that α′∈N(α), and thus {β∈Q1∣βα∈F}⊆{β∈Q1∣βα′∈F}. The following notion is taken from [12, Section 5]; compare [17]. Definition 4.4 Let A=kQ/I be a quadratic monomial algebra. Denote by F the set consisting of paths in Q, that are of length two and contained in I. The relation quiver RA of A is defined as follows. Its vertices are given by arrows in Q, and there is an arrow [βα] from α to β for each element βα in F. We will consider the algebra B=kRA/J2 with radical square zero defined by RA. The main result of this paper is as follows. Theorem 4.5 Let A=kQ/Ibe a quadratic monomial algebra, and let B=kRA/J2be the algebra with radical square zero defined by the relation quiver of A. Then there is a pre-triangle quasi-equivalence connecting A-mod̲and B-mod̲. Consequently, there is a singular equivalence between Aand B. For an arrow α in Q, we denote by Sα and Pα the simple B-module and the indecomposable projective B-module corresponding to the vertex α, respectively. We may identify Pα with Beα, where eα denotes the trivial path in RA at α. Hence, the B-module Pα has a k-basis {eα,[βα]∣βα∈F}. We observe the following short exact sequence of B-modules 0⟶⨁{β∈Q1∣βα∈F}Sβ⟶iαPα⟶Sα⟶0, (4.3) where iα identifies Sβ with the B-submodule k[βα]. We denote by B-ssmod̲ the full subcategory of B-mod̲ consisting of semisimple B-modules. We observe that for any B-module M, the syzygy ΩB(M) is semisimple; compare [11, Lemma 2.1]. Moreover, any homomorphism f:X→Y between semisimple modules splits, that is, it is isomorphic to a homomorphism of the form (00IdZ0):K⊕Z→C⊕Z for some B-modules K, C and Z. We infer that B-ssmod̲⊆B-mod̲ is a left triangulated subcategory. Moreover, all left triangles inside B-ssmod̲ are direct sums of trivial ones. There is a unique k-linear functor F:B-ssmod̲→A-mod̲ sending Sα to Aα for each arrow α in Q. Here, for the well-definedness of F, we use the following fact, which can be obtained by comparing (4.1) and (4.3): the simple B-module Sα is projective if and only if so is the A-module Aα. We have the following key observation. Lemma 4.6 The functor F:B-ssmod̲→A-mod̲is a pre-triangle equivalence. Proof Let α be an arrow in Q. We observe that (4.1) and (4.3) compute the syzygies modules ΩA(Aα) and ΩB(Sα), respectively. It follows that the functor F commutes with the syzygy functors. In other words, there is a natural isomorphism δ:FΩA→ΩBF such that (F,δ) is a looped functor. Since all morphisms in B-ssmod̲ split, each left triangle inside is a direct sum of trivial ones. It follows that F respects left triangles, that is, (F,δ) is a triangle functor. We verify the conditions (S1)–(S3) in Proposition 2.2. Then we are done. Since the functor F is faithful, (S2) follows. The condition (S3) follows from Lemma 4.1(3). For (S1), we take a morphism g:F(X)→F(Y) in A-mod̲. Without loss of generality, we assume that both X and Y are indecomposable, in which case both are simple B-modules. We assume that X=Sα and Y=Sα′. We assume that g is non-zero, in particular, F(X)=Aα is non-projective, or equivalently, the set {β∈Q1∣βα∈F} is non-empty. Observe that F(Y)=Aα′. We apply Lemma 4.2(3) to infer that α′∈N(α). Write π=πα,α′. By Lemma 4.2(4), we may assume that g=π¯. The commutative diagram in Remark 4.3 implies that ΩA(g) equals the inclusion morphism ⨁{β∈Q1∣βα∈F}Aβ⟶⨁{β∈Q1∣βα′∈F}Aβ. Take f to be the corresponding inclusion morphism ΩB(Sα)=⨁{β∈Q1∣βα∈F}Sβ⟶ΩB(Sα′)=⨁{β∈Q1∣βα′∈F}Sβ in B-ssmod̲. Then we identify F(f) with ΩA(g); more precisely, we have F(f)=δY◦ΩA(g)◦(δX)−1. This proves the condition (S1).□ We now prove Theorem 4.5. Proof of Theorem4.5. Consider the inclusion functor inc:B-ssmod̲→B-mod̲. As mentioned above, this is a triangle functor. Recall that the syzygy of any B-module is semisimple, that is, it lies in B-ssmod̲. Then the inclusion functor is a pre-triangle equivalence by Corollary 2.3. Recall the pre-triangle equivalence F:B-ssmod̲→A-mod̲ in Lemma 4.6. Then we have the required pre-triangle quasi-equivalence A-mod̲⟵FB-ssmod̲⟶incB-mod̲. The last statement follows from Proposition 3.2. We mention that by the explicit construction of the functor F, the resulting triangle equivalence Dsg(A)→Dsg(B) sends Aα to Sα for each arrow α in Q.□ Remark 4.7 We will observe in the proof of Proposition 5.3 below that the singular equivalence in Theorem 4.5 restricts to a triangle equivalence between A-Gproj̲ and B-Gproj̲. Consequently, it induces a triangle equivalence between Ddef(A) and Ddef(B). We emphasize that in general a singular equivalence will not induce a triangle equivalence between Gorenstein defect categories. 5. Consequences and examples In this section, we draw some consequences of Theorem 4.5 and describe some examples. We first make some preparation by recalling some known results on the singularity category of an algebra with radical square zero. For a finite quiver Q, we recall that a vertex in Q is a sink if there is no arrow starting at it. We denote by Q0 the quiver without sinks, that is obtained from Q by repeatedly removing sinks. The double quiver Q¯ of Q is obtained from Q by adding for each α∈Q1 a new arrow α* in the reverse direction, that is, s(α*)=t(α) and t(α*)=s(α). Recall that the Leavitt path algebra L(Q) of Q with coefficients in k is the quotient algebra of kQ¯ modulo the two-sided ideal generated by the following elements: {αβ*−δα,βet(α)∣α,β∈Q1}∪{∑{α∈Q1∣s(α)=i}α*α−ei∣i∈Q0non-sink}. Here, δ denotes the Kronecker symbol. Then L(Q) has a natural Z-grading by degei=0degα=1 and degα*=−1. We denote by L(Q)-grproj the category of finitely generated Z-graded left L(Q)-modules, and by (−1):L(Q)-grproj→L(Q)-grproj the degree-shift functor by degree −1. For details on Leavitt path algebras, we refer to [1, 13, 27]. We denote by kQ/J2 the algebra with radical square zero defined by Q. For n≥1, we denote by Zn the basic n-cycle, which is a connected quiver consisting of n vertices and n arrows which form an oriented cycle. Then the algebra kZn/J2 is self-injective. In particular, the stable module category kZn/J2-mod̲ is triangle equivalent to Dsg(kZn/J2). An abelian category A is semisimple if any short exact sequence splits. For example, if the quiver Q has no sinks, the category L(Q)-grproj is a semisimple abelian category; see [13, Lemma 4.1]. For a semisimple abelian category A and an auto-equivalence Σ on A, there is a unique triangulated structure on A with Σ the translation functor. Indeed, all triangles are direct sums of trivial ones. The resulting triangulated category is denoted by (A,Σ); see [10, Lemma 3.4]. As an example, we will consider the triangulated category (L(Q)-grproj,(−1)) for a quiver Q without sinks. Example 5.1 Let kn=k×k×⋯×k be the product algebra of n copies of k. Consider the automorphism σ:kn→kn sending (a1,a2,…,an) to (a2,…,an,a1), which induces an automorphism σ*:kn-mod→kn-mod by twisting the kn-action on modules. We observe that there are triangle equivalences (kn-mod,σ*)⟶∼kZn/J2-mod̲⟶∼(L(Zn)-grproj,(−1)). The first equivalence is well known and the second one is a special case of [13, Theorem 6.1]. We will denote this triangulated category by Tn. Let Q be a finite quiver. We call a connected component C of Qperfect (resp. acyclic) if it is a basic cycle (resp. it has no oriented cycles). A connected component is defect if it is neither perfect nor acyclic. Then we have a disjoint union Q=Qperf∪Qac∪Qdef, where Qperf (resp. Qac, Qdef) is the union of all the perfect (resp. acyclic, defect) components in Q. Denote by B=kQ/J2. Then we have a decomposition of algebras B=Bperf×Bac×Bdef. We summarize the known results on the singularity category and the Gorenstein defect category of an algebra with radical square zero. Lemma 5.2 Keep the notation as above. Then the following statements hold: There is a triangle equivalence Dsg(B)≃Bperf-mod̲×Dsg(Bdef). There is a triangle equivalence B-Gproj̲≃Bperf-mod̲, which is triangle equivalent to a product of categories Tn. There is a triangle equivalence Ddef(B)≃Dsg(Bdef), which is triangle equivalent to (L((Qdef)0)-grproj,(−1)). Proof We observe that the algebra Bperf is self-injective and that Bac has finite global dimension. Then (1) is a consequence of the decomposition Dsg(B)=Dsg(Bperf)×Dsg(Bac)×Dsg(Bdef) of categories. For (2), we note that any Bperf-module is Gorenstein-projective and that a Gorenstein-projective Bac-module is necessarily projective. By [11, Theorem 1.1] any Gorenstein-projective Bdef-module is projective. Then (2) follows by a similar decomposition of B-Gproj̲. The last statement follows from Example 5.1, since Bperf is isomorphic to a product of algebras of the form kZn/J2. By (1) and (2), the functor GB:B-Gproj̲→Dsg(B) is identified with the inclusion. The required triangle equivalence in (3) follows immediately. The last sentence follows by combining [10, Proposition 4.2] and [13, Theorem 6.1]; compare [10, Theorem B] and [27, Theorem 5.9].□ In what follows, let A=kQ/I be a quadratic monomial algebra with RA its relation quiver. We denote by {C1,C2,…,Cm} the set of all the perfect components in RA, and by di the number of vertices in the basic cycle Ci. Let B=kRA/J2 be the algebra with radical square zero defined by RA. We consider the triangle equivalence Φ:Dsg(A)→Dsg(B) obtained in Theorem 4.5. We identify the fully faithful functors GA and GB as inclusions. The following result describes the singularity category and the Gorenstein defect category of a quadratic monomial algebra. We mention that the equivalence in Proposition 5.3(2) is due to [12, Theorem 5.7], which is obtained by a completely different method. Proposition 5.3 The triangle equivalence Φ:Dsg(A)→Dsg(B)restricts to a triangle equivalence A-Gproj̲→∼B-Gproj̲, and thus induces a triangle equivalence Ddef(A)→∼Ddef(B). Consequently, we have the following triangle equivalences: Dsg(A)→∼A-Gproj̲×Ddef(A); A-Gproj̲→∼Bperf-mod̲→∼Td1×Td2×⋯×Tdm; Ddef(A)→∼Dsg(Bdef)→∼(L(Q′)-grproj,(−1)) with Q′=(RAdef)0. Proof Recall from the proof of Theorem 4.5 that Φ(Aα)=Sα for each arrow α in Q. By [12, Lemma 5.4(1)] the A-module Aα is non-projective Gorenstein-projective if and only if α, as a vertex, lies in a perfect component of RA. Moreover, any indecomposable non-projective Gorenstein-projective A-module arises in this way. On the other hand, any indecomposable non-projective Gorenstein-projective B-module is of the form Sα with α in RAperf; see Lemma 5.2(2). It follows that the equivalence Φ restricts to the equivalence A-Gproj̲→∼B-Gproj̲. The three triangle equivalences follow immediately from the equivalences in Lemma 5.2.□ We end the paper with examples on Proposition 5.3. Example 5.4 Let A be a quadratic monomial algebra which is Gorenstein. By [12, Proposition 5.5(1)], this is equivalent to the condition that the relation quiver RA has no defect components. For example, a gentle algebra is such an example. Note that Ddef(A) is trivial. Then we obtain a triangle equivalence Dsg(A)⟶∼Td1×Td2×⋯×Tdm, where di’s denote the sizes of the perfect components of RA. This result extends [20, Theorem 2.5(b)]; see also [8]. Example 5.5 Let A=k⟨x,y⟩/I be the quotient algebra of the free algebra k⟨x,y⟩ by the ideal I=(x2,y2,yx). Then the relation quiver RA is as follows: The relation quiver has no perfect components. Then we have triangle equivalences Dsg(A)≃Ddef(A)≃(L(RA)-grproj,(−1)). Example 5.6 Consider the following quiver Q and the algebra A=kQ/I with I=(βα,αβ,δγ,γδ,δξ): Its relation quiver RA is as follows: There are one perfect component and one defect component; moreover, we observe (RAdef)0=Z2. Then we have triangle equivalences A-Gproj̲≃T2 and Ddef(A)≃(L(Z2)-grproj,(−1)), which is equivalent to T2; see Example 5.1. 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The Quarterly Journal of Mathematics – Oxford University Press

**Published: ** Mar 12, 2018

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